Help with solving simple equations

Solving equations

Remember, equations must always balance.

What is on one side of the equals sign = must be balanced by what is on the other side.

So, if you subtract something from one side, the same amount needs to be subtracted from the other side.

Similarly for addition, multiplication and division.

Worked examples

Example 1

P-5=12

We need to get rid of the -5 from the left-hand side of the equation to leave P on its own.

So, add 5 to either side of the equation.

Remember +5 is the inverse of -5.

(Inverse is the mathematical word for opposite.)

This gives:

P-5+5=12+5

The solution is:

P=17

Example 2

P+12=24

We need to get rid of the +12 from the left-hand side of the equation, to leave P on its own.

So, subtract 12 from both sides of the equation.

Remember, -12 is the inverse +12.

P+12-12=24-12

The solution is:

P=12

Example 3

3P=12

This is the same as:

3xP=24

We need to divide both sides by 3 to leave the P on its own.

Remember to division is the inverse operation of multiplication.

3xP all divided by 3 =12÷3

Or, written another way:

(3xP) ÷3 =12÷3

The solution is:

P=4

Example 4

P/4=12

P÷4=12

We need to multiply both sides of the equation by 4, to leave P on its own on the left-hand side of the equation.

Remember multiplication is the inverse of division.

Px4 all divided by 4 =12x4

The 4s cancel out on the left-hand side.

The solution is:

p=48

Example 5

Note: the process will not always lead to a whole number or a positive number. The answer can be a fraction or a negative number &mdash: or even both together.

For instance:

3P=4

As in the previous example, divide both sides by 3

P=4÷3

P+24=12

We need to subtract 24 from both sides of the equation to leave P on its own on the left-hand side of the equation.

P+24-24=12-24

The solution is:

P=-12

Example 6

The above examples are simple. But some equations can be a little more difficult and involve all four operators, and end with solutions that are fractions or negative numbers, and even both. The process is longer, but is the same… at each stage the equation must be kept balanced.

Take it one step at a time.

For instance:

(7W/5)+11=3

Or,

7W divided by 5 all plus 11 =3

This looks complicated, but take it a step-at-a-time. And remember to write each step on its own line.

Our aim is to get the W on its own on the left-hand side of the equation.

Let’s get rid of the 11 first by subtracting 11 from both sides of the equation.

Remember, subtraction is the inverse of plus.

(7W÷5)+11-11=3-11

Simplifying this gives:

(7W÷5)=-8

Now let’s get rid of the denominator (the number on the bottom of the division) 5. We do this by multiplying both sides by 5. Multiplication is the inverse of division.

(7Wx5÷5)=(-8)x5

Simplifying this gives:

7W=-40

Almost there.

We now need to get rid of the 7 in front of the W. To do this we divide both sides by 7:

W=-40÷7

W=-40/7

Easy! Well, if not... take a look again. Break the problem down into a series of simple one-at-a-time balancing actions.

Don’t be frightened of the solution.

You can leave the solution as a fraction.